# Projective module

A **projective module** is a module \( P \) over a ring \( R \) that behaves like a free module with respect to lifting homomorphisms: for every epimorphism \( \alpha : B \to C \) of modules and every homomorphism \( \beta : P \to C \), there is a homomorphism \( \gamma : P \to B \) with \( \beta = \alpha \gamma \).<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup> The concept, central to abstract algebra and homological algebra, generalizes the idea of a free module, that is, a module with a basis.<sup>[2](https://academickids.com/encyclopedia/index.php/Projective_module)</sup>

| Key fact | Detail |
|---|---|
| Equivalent definitions | Lifting property; direct summand of a free module; \( \mathrm{Hom}(P, -) \) is an exact functor; every epimorphism \( A \to P \) splits<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup> |
| Free modules | Every free module is projective, but the converse fails in general<sup>[3](https://mathworld.wolfram.com/ProjectiveModule.html)</sup> |
| Kaplansky's theorem (1958) | Every projective module is a direct sum of projective modules with countably many generators, reducing the structure theory to the countable case<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup> |
| When projective = free | Proved for local rings and for polynomial rings in several variables over a field (Quillen and Suslin, 1976)<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup> |
| Finite projective modules | Finitely presented and flat, equivalently a direct summand of a finite free module, equivalently finitely presented with free localizations at all primes<sup>[4](https://stacks.math.columbia.edu/tag/00NV)</sup> |
| Geometry | By the Serre-Swan theorem, finitely generated projective modules over the ring of smooth functions on a compact manifold are exactly the modules of smooth sections of vector bundles<sup>[5](https://arxiv.org/abs/1302.3310)</sup> |

## Equivalent characterizations

The four standard conditions on an \( R \)-module \( P \) are equivalent.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup>

1. **Lifting property.** For any epimorphism \( \alpha : B \to C \) and any homomorphism \( \beta : P \to C \), there exists \( \gamma : P \to B \) with \( \beta = \alpha \gamma \).<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup>
2. **Direct summand of a free module.** \( P \) is a direct summand of some direct sum \( \bigoplus_I R \).<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/ProjectiveModule.html)</sup>
3. **Exactness of Hom.** The functor \( \mathrm{Hom}(P, -) \) is exact.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup>
4. **Splitting.** Any epimorphism \( A \to P \) of modules splits.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup>

The equivalence of the lifting property and the direct-summand condition is elementary. Free modules satisfy the lifting property because a lift is determined by lifting the images of basis elements; conversely, choosing a surjection \( \pi : F \to P \) from a free module \( F \) onto \( P \), the lifting property applied to the identity map of \( P \) splits \( \pi \), giving \( F \cong P \oplus \ker(\pi) \).<sup>[6](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf)</sup> A direct summand of a projective module is again projective, and a direct sum of projectives is projective, so the class of projectives is closed under these operations.<sup>[7](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MA/MAa/Lee%20J.%20%28ed.%29%20Commutative%20rings.%20New%20research%20%28Nova%20Science%2C%202009%29%28ISBN%209781617281921%29%28O%29%28143s%29_MAa_.pdf)</sup><sup> • </sup><sup>[6](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf)</sup>

## Relation to free modules

Every free module is projective, since a free module is a direct summand of itself. The converse does not hold in general: a projective module need not itself be a direct sum of copies of \( R \).<sup>[3](https://mathworld.wolfram.com/ProjectiveModule.html)</sup>

Concrete examples show the gap. Over \( R = \mathbb{Z}/6\mathbb{Z} \), the module \( P = \mathbb{Z}/3\mathbb{Z} \) is projective but not free; the quotient map \( \mathbb{Z}/6\mathbb{Z} \to \mathbb{Z}/3\mathbb{Z} \) splits. The same module \( \mathbb{Z}/3\mathbb{Z} \) is not projective as a \( \mathbb{Z} \)-module, because the quotient map \( \mathbb{Z} \to \mathbb{Z}/3\mathbb{Z} \) does not split; in fact every projective \( \mathbb{Z} \)-module is free.<sup>[8](https://math.ucla.edu/~sharifi/notes/algebra-ch12.html)</sup> Another example is \( M = \mathbb{Z} \) regarded as a module over \( R = \mathbb{Z} \oplus \mathbb{Z} \) with multiplication \( (a \oplus b) \cdot x = ax \); \( M \) is a direct summand of the free module \( R \) but is not itself free over \( R \).<sup>[3](https://mathworld.wolfram.com/ProjectiveModule.html)</sup> For \( n \ge 2 \), the module of column vectors over the matrix ring \( M_n(R) \) is projective, being a direct summand of \( M_n(R) \) itself, but is not free.<sup>[8](https://math.ucla.edu/~sharifi/notes/algebra-ch12.html)</sup> Nonprojective modules also exist among familiar ones: \( \mathbb{Q} \) is a nonprojective \( \mathbb{Z} \)-module, since it is not free.<sup>[3](https://mathworld.wolfram.com/ProjectiveModule.html)</sup>

Over certain classes of rings the two notions do coincide. The coincidence has been proved for local rings, and for rings of polynomials in several variables over a field.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup> Over a principal ideal domain, every projective module is free.<sup>[8](https://math.ucla.edu/~sharifi/notes/algebra-ch12.html)</sup> Over rings decomposable into a direct sum, there always exist projective modules different from free ones.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup>

## Finitely generated projective modules

For finitely generated modules the theory is especially clean. The Stacks project records the following equivalences for an \( R \)-module \( M \): \( M \) is finitely presented and flat; \( M \) is finite projective; \( M \) is a direct summand of a finite free module; \( M \) is finitely presented and \( M_{\mathfrak p} \) is free for every prime \( \mathfrak p \); and \( M \) is finitely presented and \( M_{\mathfrak m} \) is free for every maximal ideal \( \mathfrak m \).<sup>[4](https://stacks.math.columbia.edu/tag/00NV)</sup> In particular, being finite projective over a local ring is the same thing as being finite free.<sup>[4](https://stacks.math.columbia.edu/tag/00NV)</sup> A finitely generated flat module over a local ring is free.<sup>[9](https://arxiv.org/abs/1701.07735)</sup>

If \( P \) is projective and generated by \( n \) elements, then there is a decomposition \( P \oplus Q \cong R^n \): the generators give a surjection \( \pi : R^n \to P \), and the lifting property splits it.<sup>[6](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf)</sup> Finitely generated projective modules can also be described by idempotent matrices: \( P \) is projective of finite type over \( A \) if and only if there is an idempotent \( e \in M_N(A) \), meaning \( e^2 = e \), with \( P \cong eA^N \).<sup>[10](https://www.sciencedirect.com/topics/mathematics/projective-module)</sup>

Over an integral domain, an ideal \( I \) is projective if and only if it is invertible, and such an ideal is then finitely generated.<sup>[10](https://www.sciencedirect.com/topics/mathematics/projective-module)</sup> A ring in which every ideal is projective is called hereditary; every \( R \)-module is projective if and only if \( R \) is a finite product of fields.<sup>[10](https://www.sciencedirect.com/topics/mathematics/projective-module)</sup>

## Kaplansky's theorem and structure

Kaplansky's theorem, proved in his 1958 paper "Projective modules" in the *Annals of Mathematics*, asserts that every projective module is a direct sum of projective modules with countably many generators, reducing the study of the structure of projective modules to the countable case.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup> Projective modules with finitely many generators are studied in algebraic \( K \)-theory.<sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup>

## The Quillen-Suslin theorem

In his 1955 paper *Faisceaux algébriques cohérents*, [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre), a French mathematician noted for foundational work in algebraic topology and algebraic geometry, remarked "On ignore s'il existe des \( K[x_1, \ldots, x_n] \)-modules projectifs de type fini qui ne soient pas libres", asking whether finitely generated projective modules over a polynomial ring over a field must be free.<sup>[11](https://arxiv.org/abs/2112.02272)</sup> In 1976 [Daniel Quillen](https://www.edgepedia.org/daniel-quillen), in "Projective modules over polynomial rings" (*Inventiones Mathematicae* **36**, pp. 167-171), and [Andrei Suslin](https://www.edgepedia.org/andrei-suslin), in "Projective modules over a polynomial ring are free" (*Soviet Math. Dokl.* **17**:4, pp. 1160-1164), independently proved that every finitely generated projective \( K[x_1, \ldots, x_n] \)-module is free; the result is now known as the Quillen-Suslin theorem.<sup>[12](https://bishtref.com/articles/10.1007/bf01390008)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Projective_module)</sup><sup> • </sup><sup>[11](https://arxiv.org/abs/2112.02272)</sup> Quillen's proof proceeds through a localization theorem and an extension theorem, and a 1977 seminar write-up by Warren Dicks, a mathematician at the Universitat Autònoma de Barcelona, recasts the argument in the language of idempotent matrices.<sup>[11](https://arxiv.org/abs/2112.02272)</sup>

## Projective modules and vector bundles

Projective modules carry a geometric interpretation. In differential geometry, the Serre-Swan theorem gives an equivalence of categories between smooth vector bundles on a smooth compact manifold \( \mathcal{M} \) and finitely generated projective modules over the ring \( C^{\infty}(\mathcal{M}) \) of smooth functions; a bundle \( E \to \mathcal{M} \) corresponds to the module \( \Gamma^{\infty}(E) \) of its smooth sections.<sup>[5](https://arxiv.org/abs/1302.3310)</sup> Serre had already connected the two notions in his 1957/58 seminar talk "Modules projectifs et espaces fibrés à fibre vectorielle".<sup>[13](https://www.numdam.org/item/?id=SD_1960-1961__14_1_A2_0)</sup> In the topological setting, a theorem of Swan states that the global section functor is an equivalence from the category of vector bundles over a [compact space](https://www.edgepedia.org/compact-space) \( X \) to the category of finitely generated projective modules over the ring \( k(X) \) of continuous functions.<sup>[14](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/bass.pdf)</sup>

The same finitely generated projective modules organize algebraic \( K \)-theory. The category \( \mathbf{P}(R) \) of finitely generated projective \( R \)-modules is an additive category, since the direct sum of projectives is projective, and it is the input to the Grothendieck group \( K_0 \) of a ring.<sup>[6](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf)</sup><sup> • </sup><sup>[15](https://sites.math.rutgers.edu/~weibel/Kbook.html)</sup>

## Role in homological algebra

Projective modules are one of the three fundamental classes, alongside injective and flat modules, in basic homological algebra.<sup>[16](https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/gorenstein-projective-injective-and-flat-modules/D95889FF31F4988B1EAE7FE6A734E71A)</sup> A module has finite projective dimension when it admits a finite resolution by projectives, and this invariant measures the ring itself: by the Auslander-Buchsbaum-Serre theorem, a local [Noetherian ring](https://www.edgepedia.org/noetherian-ring) is regular if and only if every finitely generated module has finite projective dimension.<sup>[17](https://math.ug/alggeom3-ws2324/sect0002.html)</sup> For a finitely generated module \( M \) of finite projective dimension over a local Noetherian ring \( A \), the Auslander-Buchsbaum formula reads \[ \operatorname{depth}(A) = \operatorname{projdim}_A(M) + \operatorname{depth}_A(M). \]<sup>[17](https://math.ug/alggeom3-ws2324/sect0002.html)</sup>

Projectivity is stronger than flatness. A finite module that is flat need not be projective,<sup>[18](https://stacks.math.columbia.edu/download/book.pdf)</sup> and in general there exist finitely generated flat modules that are not projective.<sup>[9](https://arxiv.org/abs/1701.07735)</sup> The two notions agree for finitely presented modules over any ring.<sup>[4](https://stacks.math.columbia.edu/tag/00NV)</sup>

## References

1. [Projective module, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Projective_module)
2. [Projective module, Academic Kids encyclopedia entry](https://academickids.com/encyclopedia/index.php/Projective_module)
3. [Projective Module, Wolfram MathWorld](https://mathworld.wolfram.com/ProjectiveModule.html)
4. [Section 10.78: Finite projective modules, The Stacks Project](https://stacks.math.columbia.edu/tag/00NV)
5. [A Serre-Swan theorem for bundles of bounded geometry, arXiv:1302.3310](https://arxiv.org/abs/1302.3310)
6. [The K-book: an introduction to algebraic K-theory, Charles A. Weibel, Chapter I](https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.pdf)
7. [Commutative Rings: New Research, Nova Science, excerpt](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MA/MAa/Lee%20J.%20%28ed.%29%20Commutative%20rings.%20New%20research%20%28Nova%20Science%2C%202009%29%28ISBN%209781617281921%29%28O%29%28143s%29_MAa_.pdf)
8. [Abstract Algebra, Chapter 12: Homological algebra, Romyar Sharifi, UCLA notes](https://math.ucla.edu/~sharifi/notes/algebra-ch12.html)
9. [On the projectivity of finitely generated flat modules, Abolfazl Tarizadeh, arXiv:1701.07735](https://arxiv.org/abs/1701.07735)
10. [Projective Module - an overview, ScienceDirect Topics (Handbook of Algebra; Handbook of Global Analysis)](https://www.sciencedirect.com/topics/mathematics/projective-module)
11. [Seminar on Quillen's proof of the Quillen-Suslin theorem, Warren Dicks, arXiv:2112.02272](https://arxiv.org/abs/2112.02272)
12. [Projective modules over polynomial rings, Daniel Quillen, Inventiones Mathematicae 36 (1976), pp. 167-171](https://bishtref.com/articles/10.1007/bf01390008)
13. [Sur les modules projectifs, Séminaire Dubreil-Pisot (Numdam), bibliography](https://www.numdam.org/item/?id=SD_1960-1961__14_1_A2_0)
14. [Appendix on Swan's theorem, book by Hyman Bass (excerpt)](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/bass.pdf)
15. [The K-book: an introduction to algebraic K-theory, Charles A. Weibel, table of contents](https://sites.math.rutgers.edu/~weibel/Kbook.html)
16. [Gorenstein projective, injective and flat modules, Journal of the Australian Mathematical Society](https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/gorenstein-projective-injective-and-flat-modules/D95889FF31F4988B1EAE7FE6A734E71A)
17. [Algebraic Geometry 3 lecture notes: Serre duality, Auslander-Buchsbaum formula](https://math.ug/alggeom3-ws2324/sect0002.html)
18. [The Stacks Project, chapter on finite projective modules (PDF)](https://stacks.math.columbia.edu/download/book.pdf)

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